#27193206

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₁₀₃ ⋊ #1366

Subgroups

	subgroup	G/H
center	C₂	#20267539
commutator	C₄²×C₁₀₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₃
Fitting	#364×C₁₀₃	C₂
socle	C₂×C₁₀₃	#333
prime radical	C₂²×C₁₀₃	#100
cosocle	C₄×C₈×C₁₀₃	C₂²
hypercenter	#364	D₁₀₃
hypocenter	C₁₀₃	#1366
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 3296

A minimal faithful permutation representation — 2 generators on 135 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(104,106,109,116)(105,110)(107,119,122,124)(108,123,120,118)(111,115)(112,128)(113,132)(114,126)(117,121,133,131)(125,129)(127,134)(130,135);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105,107,112,120,113,131,129,109,114,122,130,108,111,121,127)(106,115,117,128,123,126,119,134,116,132,133,135,118,110,124,125)

Polycyclic presentation:

2^7,103|a^2=d,b^2=f,c^2=fg,d^2=ef,e^2=g,b^a=bc,c^a=ce,e^a=eg,f^a=fg,h^a=h^102,c^b=cf,d^b=def,e^b=eg,h^b=h^102

Conjugacy classes & character table

1451 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 2572);